This thread is the based on items developed in many threads on the Atlas Forum and except for erosity was initially codified by the Forum Redistricting Commission of 2014 to guide the submission and judging of plans for VA. The posts have been edited to reflect updates since that time. In that same spirit, I may make further edits based on comments and examples that arise.

Item 1. Choice of software and format. Plans shall be developed with Dave's Redistricting App using 2010 census data with Voting Age Population and City/Town lines enabled. For each district a plan shall list the Deviation (see item 7 for the definition), minority Voting Age Population (VAP) in percent for districts where it exceeds 25%, and the President 2008 results in percent. Plans must include a statewide map with City/Town lines off, and zoom views for areas where the division of counties is difficult to see from the statewide view with City/Town lines on. [Exceptions to the submission format may be granted by a majority vote of the commission.]

Item 2. Application of the Voting Rights Act. Each submitted plan must comply with the VRA. Section 2 of the VRA requires that minorities must have the opportunity to elect their candidate of choice. SCOTUS decisions have clarified that section 2 is mandated when a minority makes up at least 50%+1 of the VAP (or CVAP for Hispanics) in a compact area and there is evidence of racial bloc voting. The district does not need to have 50% of a minority to elect a candidate of choice if it can be shown that the minority is likely to prevail with a lower percentage. A district cannot be drawn where race is the predominant factor guiding the shape of the district.

Item 3. Criteria to evaluate plans - muon2's SPICE system. Plans shall be evaluated with five measures: two political measures, one demographic measure, and two geographic measures as follows.

Political measures

SKEW: a measure of the amount that a redistricting plan favors one party over another compared to the natural leanings of the state.

POLARIZATION: a measure of the number of districts that are uncompetitive for one of the major parties.

Demographic measures

INEQUALITY: a measure of the amount of population variance among districts.

Geographic measures

CHOP: a measure of the number of geographic communities of interest, such as political subdivisions, divided by separate districts.

EROSITY: a measure of the amount of irregularity of district boundaries based on the separation of connected population centers into different districts.

Use of measures

Measures are set up so that low scores are more desirable.

Item 3. Use of SPICE sores. Political measures (SKEW and POLARIZATION) may be used to judge between plans, but are not used to invalidate them. A plan is discarded if the INEQUALITY exceeds the maximum allowed by law. A combination of INEQUALITY, CHOP and EROSITY is used to eliminate plans according to one of the following methods. Multiple plans may be acceptable.

Item 3a. A plan is discarded if another plan has a lower CHOP while not increasing EROSITY. A plan is discarded if another plan has a lower EROSITY while not increasing CHOP. If two plans have equal CHOP and EROSITY, but differ on INEQUALITY the plan with higher INEQUALITY is discarded.

Item 3b. A plan is discarded if another plan has a lower CHOP+INEQUALITY while not increasing EROSITY. A plan is discarded if another plan has a lower EROSITY while not increasing the total CHOP+INEQUALITY.

These next two items will deal with the political measures for a plan. Under Item 3, they will only be used to guide the commission when they consider final plans, and can't be used to eliminate plans prior to the end of the submission period. First I will provide some definitions.

Definition: PVI. The Partisan Voting Index is based on the method developed by the Cook Political Report and widely used in assessing the political tendencies of congressional districts. It compares the Democratic share of the two-party vote in a state or a district to the Democratic share of the national presidential vote, averaged over the last two presidential elections. Shares are multiplied by 100 to get a percent, and expressed as D+x when positive and R+(-x) when negative.

Definitions: A highly competitive district has a PVI of 0 or 1 (-0.014 to +0.014) and statistically such districts have an even chance of being one by either district. A competitive district has a PVI of 2 through 5 (-0.054 to -0.015 and +0.015 to 0.054) and statistically such districts have an 3 out of 4 chance of being held by the favored party. An uncompetitive district has a PVI of 6 or greater for either party and has better than a 9 in 10 chance of being held by the favored party. This is based on congressional results during the preceding decade.

Definition: The expected delegation difference from a state with a known PVI is equal to 50%+2*PVI, so for example a D+5 state would be expected to have a delegation of 60% Democrats. Studies (e.g. Goeddert 2014) show that for every 1% shift in the national vote share there is an average shift by 2% in the number of congressional seats. Extending that to individual states, one can predict that in a 50-50 national election, a state delegation should have a Democratic fraction equal to 50% + 2*(state PVI). The percent difference between the Democratic and Republican fractions is then 4*(state PVI). The difference between the Democratic delegation size and the Republican delegation size should be 4*(state PVI/100)*(size of the delegation), where the division by 100 is to remove the percent.

Item 4: SKEW measures the partisan fairness of a plan. Count 0 for each highly competitive district, +1 for each competitive or uncompetitive Democratic district, and -1 for each competitive or uncompetitive Republican district. Take the total for all districts in the state and subtract the expected state delegation difference. Express a negative number as a positive number in favor of the Republicans. That positive number is the SKEW score, and lower numbers are closer to the ideal partisan fairness.

Item 5: POLARIZATION measures the competitiveness of a plan. Count 0 for each highly competitive district, 1 for each competitive district, and 2 for each uncompetitive district in a plan. The total for the whole state is the POLARIZATION score, and lower numbers indicate greater competitiveness.

Definition: Quota. The quota is the total population of a state divided by the number of districts rounded to the nearest whole number.Definition: Deviation. The deviation is the difference between the population of a district and the quota. Negative numbers indicate a district that has a population that is smaller than the quota.Definition: Range. The range is the difference in population between the largest and smallest district in a plan.Definition: Average Deviation. The average deviation is the average of the absolute values of the deviations for all districts in a plan.

Background: SCOTUS has set two different standards for districts. Legislative and local districts must be substantially equal and that has been interpreted to be a range not exceeding 10% of the quota. Congressional districts must be as equal as practicable, and for some time that was assumed to mean that only exact equality would do. However, the recent WV case makes it clear that a range of up to 1% of the quota is acceptable when driven by other neutral redistricting factors. Greater than 1% might also be acceptable, but 10% would presumably not be because that is set by a different standard. It's an evolving area in the law.

Item 6. All plans for congressional districts shall have a range not exceeding 1% of the quota. All other plans shall have a range not exceeding 10% of the quota except when otherwise limited by state law.

Background: Some time ago there were some threads that tried to optimize the population equality of districts with no county splits. The result of that exercise was the following graph.

Each square represents a state. New England states used towns instead of counties, and states with counties too large for a district assumed that a whole number of counties would nest inside the large county. The more counties available per district, the closer to equality one could achieve, and the relation is logarithmic in population. The green line represents the best fit to the data. Data for average deviation can be fit as well, but the result is not substantially different other than the scale factor that has the average deviation equal to about 1/4 the range.

The average state has about 72 counties and if one divides that number into 2, 3, 4, etc. districts then one can use the fit from the data in the graph to predict a likely range. That in turn can be built into a table.

Item 7. The INEQUALITY score for a plan is found by taking the range for a plan and comparing it to the table below.

I don't totally understand item 7, could you explain it a little more please, Muon?

The basic idea is that plans get sorted by chop (+ inequality) and erosity scores to determine which will go on to the commission (Item 3a or 3b). Any plan that beats another with in one score and does at least as well on the other knocks out the higher scoring plan (like golf low scores are desired). Population inequality is either used as a tie breaker if two plans are otherwise tied or combined with the chop score. The question is whether to use inequality strictly or to recognize that two plans with similar inequality remain statistically tied. That leads to the table in Item 7.

A while back I led some threads where a number of posters put together whole county plans to see how low the inequality could go, and applied the same idea to New England states with towns. The result was the graph I posted for Item 7. The fit is best when one uses the logarithm of the range rather than the actual range.

Next I wanted to use the fit line to see how repeated divisions would likely affect the range for a single state. When I total the number of counties and non-surrounded independent cities for non-New England states without AK and HI, there are 3023 for 42 states which is 71.97 counties per state. So a hypothetical average state has 72 counties.

Now if my hypothetical state has just 2 CDs, there would be an average of 36 counties for each, and the fit line predicts that the range would be 6. That corresponds to one chop of the state. A second chop of the state into 3 CDs average 24 counties for each and a predicted best range of 93. A third chop to 4 CDs gives a range of 379. Eventually a thirteenth chop predicts a range of 7656 which exceeds 1%. I took those predicted ranges for the number of state chops and rounded them off to get the table in Item 7.

One can think of each county chop as having one less whole county CD in a state, which is equivalent to having one less chop of the state as a whole. So, the table is another way of identifying when two plans have statistically equivalent ranges that would be expected for that number of whole county CDs or whole county groups of CDs.

Definition: Geographic unit. A geographic unit is a contiguous geographic area used to build plans. States are divided into primary geographic units which cover the entire area and population of the state. Plans are built from a smallest geographic unit such as vote tabulation districts (VTD), census tracts or census blocks.

Definition: Subunit. Subunits are geographic subdivisions of a primary unit or other subunit. Examples include census defined subdivisions, townships, municipalities and unincorporated areas in an urban county, and recognized neighborhood regions in a city.

Item 8. The smallest geographic unit shall be the VTD. The primary geographic unit is the county (or parish in LA) except as identified below. Counties with populations in excess of ten times the maximum allowed range shall have identified county subunits. Subunits in excess of 10 times the allowed range may have identified subunits larger than the VTD.

Item 8a. Independent cities outside VA shall be treated as primary geographic units in their states (Baltimore, St Louis, Carson City). Independent cities in VA greater than 50K or that have annexed their entire county shall be treated as primary geographic units (Alexandria, Lynchburg, Richmond, Roanoke, Hampton, Suffolk, Chesapeake, Norfolk, Portsmouth, Virginia Beach, Newport News). Independent cities less than 50K shall be treated as subunits of the county they were created from.

Item 8b. Towns in the New England states (CT, ME, MA, NH, RI, VT) shall be the primary geographic units in their states except for counties that have unorganized areas. Counties with unorganized areas remain primary geographic units.

Item 8c. Minor county divisions (MCDs) recognized by the Census are subunits of the counties in those states not covered by Item 8b.

Item 8d. Incorporated cities and school districts in unincorporated areas are subunits of counties not otherwise covered in Items 8b or 8c.

Definition: County Cluster. A connected set of counties sharing a common demographic feature. Connected means able to travel within the cluster to all counties on public roads or ferries without leaving the cluster. The size of a county cluster is the population of the cluster divided by the quota, and rounded up to the nearest whole number.

Definition: Cover. The cover of a cluster is the number of districts including all or part of any county in the cluster. The cover score for a cluster is the difference between the cover and the size of the cluster.

Definition: Pack. The pack of a cluster is the number of districts wholly contained within counties in the cluster. The pack score for a cluster is the difference between one less than the size and the pack of the cluster.

Definition: Urban County Cluster (UCC). A UCC is a county cluster where each county is within the same Census-designated Metropolitan Statistical Area (MSA), and has either an urbanized population of 25K or more, or an urbanized population of at least 40% of the total county population.

Definition: Minority County Cluster (MCC). An MCC is a county cluster where each county has 40% or more citizen voting age population (CVAP) for the same specific minority. Note that this is a more general statement than VAP, designed to be applicable for minority groups with substantial non-citizen populations including Latinos.

Item 9: A redistricting plan should avoid excess division of county clusters. The CHOP will increase by the cover score for each cluster. The CHOP will increase by the pack score for each cluster.

More discussion of the pack and cover and a specific application to the Omaha UCC in NE can be found in this thread.

Definition: Chop. A single chop is the division of a geographic unit between two districts. A second chop divides the unit between three districts. In general the number of chops is equal to the number of districts in that unit less one.

Definition: Chop size. In units with a single chop, the size of a chop is the population of the smaller district within the unit. For districts with more than one chop, chop sizes are measured in order from the smallest populated district in the unit up to but not including the district with the largest population in the unit.

Definition: Microchop. A microchop is a chop that has a size that is 0.5% or less of the quota. Microchops do not count towards the number of chops in a unit.

Definition: Macrochop. A macrochop is one or more chops in a county that has a total size in excess of 5.0% of the quota. When a macrochop of a county occurs, the subunits of the county must be considered as if they were units as well. Note that macrochops may only apply to counties with a population of more than 10% of the quota, and must apply to counties with more than 105% of the quota.

Item 10: CHOP measures the integrity of geographic units in a plan. The CHOP score is the total of all county chops not including microchops. In counties with a macrochop, chops of county subunits are added to the CHOP score, however VTDs that span county subdivisions do not increase the CHOP score.

Principle: Each plan can be represented by a planar graph of nodes and links, where each node corresponds to a discrete area in the plan, each link corresponds to a connection between nodes, and each district corresponds to a subgraph with every node assigned to a subgraph.

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

Definition: Link. A link is a representation of a connecting path between two nodes. There is at most one link between nodes. For any type of connection, the connecting path between two nodes is considered to be the path that takes the shortest time as determined by generally available mapping software.

For example, here's a map of NC counties with a graph overlaid. The circles represent nodes for the county seats and need not be in the exact location of the county seats. The lines connecting the circles are links, and the color indicates the type of the link. As described in further posts, blue and green links are regional connections. Yellow, gold, and orange links are local connection without a regional connection. Pink links show contiguity without a local connection.

Item 10a: A macrochop of a geographic unit replaces that original unit with a set of subunits that cover all of the original unit. The original unit and its node and links are no longer considered in the plan. The subunits are treated as new units in the plan with their own nodes and links. (see Item 10.)

For example, Mecklenburg county NC is sufficiently large that it must be macrochopped to form congressional districts. The county node has been replaced by subunit nodes and links. Subunits are based on the 6 independent towns and 6 planning areas for Charlotte. Blue links are regional connections between counties. Gold links are local connections. Pink links are contiguous only.

Here's the basic rule for connections. I have three special rules that will follow.

Principle: It should be possible to travel between all areas in a district without leaving the district. There may be parts of discrete areas that are not accessible as long as the node for the area can be reached.

Definition: Local connection. There is a local connection between two nodes if there is a continuous path of public roads and ferries that allow one to travel between the two nodes without entering any other unit. Roads along the border of two units are considered to be in either or both of the units as needed to form a connection.

Item E1: Each node in a district must be able to trace a path of local connections to every other node in the district by way only of nodes in that district.

Here were the examples that were fine for the basic connection rule.

Example 1:

In this example I'll call the 5 geographic units Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes are indicated with stars and the roads are shown with heavy lines.

From the basic rule the following connections exist:

Dawes is connected to both Calhoun and Elbridge each by a single path.

Agnew is connected to both Burr and Dawes, each by two separate paths. The shortest one by time would count as the connecting path.

Burr is not connected to Calhoun. The obvious shortest path cuts a corner of Agnew and no other path stays only within those two units.

Agnew is connected to Calhoun by a single path. The shortest path cuts a corner of Dawes, so it's not a connection. There is a valid connection that dips south towards Burr first but stays only in Agnew and Calhoun.

Elbridge is connected to both Agnew and Burr by virtue of a road that runs along the boundary of Agnew and Burr.

The equivalent graph reduces each connection to a single link between nodes.

Example 2:

In this example there are 4 geographic units: Adlai, Bryan, Clay, and Dewey. As before the nodes are indicated with stars and the roads are shown with heavy lines. The thick shaded area running roughly vertical represents a natural barrier such as a river. Think of Dewey as an independent city that has grown along the river annexing land in Adlai.

Adlai, Bryan and Dewey are all mutually connected to each other.

Clay is connected to Bryan, but not to Adlai or Dewey. The path from Clay to Dewey must go through either Adlai or Bryan. One path from Clay to Adlai initially goes into Adlai but then goes through Dewey before reaching the node of Adlai.

Here's the equivalent graph. Note that on the graph the nodes need not be in their actual location. It's the relative connection of nodes that matters.

The first special rule for connections is for fragments. This follows what was worked out earlier in the thread.

Definition: Fragment. A fragment is the contiguous part of a unit entirely within a district formed by a chop of that unit. Fragments are artifacts of a specific redistricting plan and need not correspond to a recognized political unit. The node of the fragment containing the node of the chopped unit is that same node. For a fragment that does not contain the node of the whole political unit, the node is that of the most populous subunit in the fragment.

Item E2: Fragments trace paths to their nodes as if they were part of the original unit. A connection to the node of a fragment exists if the connecting path to the unit with the fragment enters the unit in that fragment. Fragments within the same unit are locally connected if their nodes are locally connected.

This example is based on the the previous unchopped one. The 5 geographic units are Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes are indicated with stars and the roads are shown with heavy lines.

The shaded area represents a district that chops unit Agnew.

The East Agnew fragment has a node from the original whole Agnew. The West Agnew fragment has a newly created node shown as a hollow star that will be used as a placeholder for mapping.

The path from Agnew (before the chop) to Calhoun without a chop enters Agnew in the West Agnew fragment, so there is a link from West Agnew to Calhoun.

The path from Agnew to Elbridge enters Agnew in the East Agnew fragment, so there is a link from East Agnew to Elbridge.

The primary path from Agnew to Dawes enters in East Agnew, so there is a link from East Agnew to Dawes. A secondary path from Agnew to Dawes enters in West Agnew, but does not form a link.

The primary path from Agnew to Burr enters in East Agnew, so there is a link from East Agnew to Burr. A secondary path from Agnew to Burr enters in West Agnew, but does not form a link.

There is a path between West Agnew and East Agnew that forms a local connection, so there is a link between those fragments.

The equivalent graph colors the nodes of the two districts in different colors. The dashed lines represent secondary paths between nodes that do not count as links. The red lines indicate links and secondary paths that link nodes in different districts.

nb. As noted in the example the dashed lines represent real local connections, but not links. Links are based on the shortest connecting path. That's important in cases where a path of local connections is required, but no link is available. This situation will be addressed further in the special rule for isolated fragments and in the definition for components.

The second of the special rules deals with a situation that arises in many circumstances from independent cities in VA to cities embedded in townships in OH. I'm calling these situations shared units. The treatment of shared units mirrors the language for fragments within a unit which should make it easier to apply.

Definition: Shared units. Shared units occur when a unit is surrounded or effectively surrounded by another unit, making both the surrounded and surrounding units shared units. A unit is effectively surrounded when it only has local connections to the surrounding unit or other units surrounded by the same unit. Shared units also exist if the node of a unit is in another unit.

Item E3: Shared units trace paths to their nodes as if they were a single unit. A connection to the node of a specific unit among the shared units exists if the connecting path from another node enters the shared units in that specific unit. Units within a set of shared units are locally connected if their nodes are locally connected. (nb. Shared units are often a city that formed a separate unit from its surrounding county or township, but the roads are designed to go to and through the city. The city should not act as a barrier for connection in those cases.)

Example 1:

Louisville OH (yellow) is an incorporated city in Nimishillen township (blue). Both are subunits of Stark county. The node for Louisville is shown with a white star, and the node for Nimishillen is shown with a blue star. Nimishillen township is bordered by four townships in each of the cardinal directions, and the city of Canton in the southwest (outlined in pink).

Under the basic rule for connections, every path from the node of Nimishillen to any neighbor would have to go through Louisville since the node is in Louisville. That would leave Nimishillen unconnected to any neighbor.

Louisville is surrounded by Nimishillen and contains the node for Nimishillen. That makes Louisville and Nimishillen shared units both ways under the special rule. Paths to both nodes are considered using the shared units together.

All paths to either node enter the shared unit in Nimishillen. So, Nimishillen is connected to all of the neighboring townships and Canton city. Louisville is only connected to Nimishillen.

nb. It is insufficient to define the shared unit only by considering that Louisville is surrounded. Suppose that Louisville annexed more land to the west and became directly connected to Canton city. It would not be surrounded, but the node of Nimishillen would still be inside of Louisville. The special rule would take care of that situation as well.

Example 2:

East Sparta OH (pink outline) is an incorporated city in Pike township, both shown in green. Both East Sparta city and Pike township are subunits of Stark county. Their nodes are shown by stars.

Tuscarawas county in shown in blue. There are no local roads connecting East Sparta to Tuscarawas, and no path from the node of Pike to Tuscarawas. With just the basic connection rule, neither East Sparta nor Pike have connections to Tuscarawas, despite the presence of OH-800 going from the township to Tuscarawas.

East Sparta is only connected to Pike, so it is effectively surrounded. That makes Pike and East Sparta shared units.

By the special rule for shared units, we look at paths for both East Sparta and Pike as they go through the combined shared unit. OH-800 is the shortest path to the nodes of both East Sparta and Pike. It enters the combined shared unit in Pike. By the special rule Pike is connected to Tuscarawas. East Sparta remains only connected to Pike.

The third and final special rule for local connections is another one we have looked at earlier in this thread.

Definition: Isolated unit or fragment. An isolated unit has no connections from its node to the nodes of any other units based on the basic connection rule or the special rule for shared units. An isolated fragment has no connections based on the special rule for fragments.

Item E4: An isolated unit has a local connection to another unit if there is a local connection from its node to the node of any contiguous unit or subunit in that order of priority. Subunits can be further divided until at least one local connection is established. (nb. This is a fall back when the normal rules leave a unit with no local connections. There should always be a way to get to the population from outside the unit, even if it is by way of other parts of units.)

Example 1:

In this example there are 4 geographic units: Adlai, Bryan, Clay, and Dewey. As before the nodes are indicated with stars and the roads are shown with heavy lines. The thick shaded area running roughly vertical represents a natural barrier such as a river. Think of Dewey as an independent city that has grown along the river annexing land in Adlai.

In the example for the basic rule, Clay is connected to Bryan, but not to Adlai or Dewey. One path from Clay to Adlai initially goes into Adlai but then goes through Dewey before reaching the node of Adlai so it fails the basic connection rule.

In this case a chop follows the river splitting Adlai into east and west fragments. The east fragment has the node for Adlai. The west fragment (shaded) has a node based on its population shown by a hollow star.

There is no road that directly connects the east and west fragments so there is no local connection between the fragments. Since there is no connection from Clay to Adlai the road that cuts through the northern part of the West Adlai fragment also doesn't become a link to West Adlai. That makes West Adlai an isolated fragment.

Under the special rule for isolated fragments the node in West Adlai is used with its roads to determine connections. West Adlai has a path to both Clay and Dewey, so it becomes connected to both.

The equivalent graph colors the nodes of the two districts in different colors. The blue lines represent links between nodes in the same district. The red lines indicate links between nodes in different districts.

Example 2:

This is an example from the area around Canton OH. The colors represent different subunits of Stark county and the nodes are indicated by stars.Canton city (white)Canton township (green)Plain township (dark blue)North Canton city (light blue, actual location slightly north of image)Meyers Lake village (red)

Both Canton and Plain townships are split into discontiguous parts by Canton city. As long as those townships are not chopped they are viewed each as single units. Canton city and the townships have sufficient other connections and do not meet the definition of shared units.

Meyers Lake is only bordered by the two townships, and those are both discontiguous from the parts with the nodes. Meyers Lake has no path to any other node so it is an isolated unit.

Based on the special rule Meyers Lake would consider counties and then townships for connections, but none would exist. The rule allows the subunits to be further divided, in this case to precincts. Meyers Lake has connections to precincts in both townships, so Meyers Lake has local connections to both townships by the isolated unit rule.

The previous posts defined the rules for local connections which are required for all areas within a district. This post defines regional connections and what constitutes a link for erosity.

Principle: Road connections between areas are an indication of a community of interest between those areas. State and federal highways are a stronger indication of a connection than local roads. Large areas at the regional scale of counties should be connected by better roads than small areas at the scale of smaller governmental units.

Definition: Regional connection. There is regional connection if there is a continuous path of all season numbered state or federal highways or regularly scheduled ferries that allow one to travel between the nodes of two geographic units without entering any other geographic unit. If the node is not on a numbered highway, then the connection is measured from the point of the nearest numbered highway in the geographic unit to the node. Highways along the border of two units are considered to be in either or both of the units as needed to form a connection.

Item E6: A link exists between two nodes in the same county if there is a local connection between the nodes. A link exists between two nodes in different counties if there is a regional connection between nodes.

Example:

This example is based on the one for connection. The 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. There are are three districts creating three fragments from Agnew, called West Agnew, Central Agnew, and East Agnew. The nodes for the counties are indicated with solid stars and the nodes for the fragments are shown with hollow stars.

Roads that count for regional connections are shown with heavy brown lines. They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.

Since the regional roads are in the same configuration and they follow the same rules as local roads, these are the regional connections:

Burr is regionally connected to Central Agnew. The regional path from Burr to Agnew enters Agnew in the Central Agnew fragment.

Burr is only locally connected to Calhoun. The obvious shortest regional path cuts a corner of Agnew and no other path stays only within those two units.

Burr is regionally connected to Elbridge. The regional path is along the border between Burr and Agnew which counts for staying in Burr.

Calhoun is regionally connected to Central Agnew. The regional path from Calhoun to Agnew enters Agnew in the Central Agnew fragment.

Calhoun is only locally connected to West Agnew. The regional path entered at Central Agnew, and though it passes through West Agnew, that does not count as a connection for a fragment.

Calhoun is regionally connected to Dawes.

Dawes is regionally connected to Central Agnew. The shortest regional path from Dawes to Agnew enters Agnew in the Central Agnew fragment.

Dawes is regionally connected to West Agnew. Neither shortest regional path from Calhoun or Dawes entered at West Agnew, so it is regionally isolated. As an isolated fragment it uses other regional links, and that includes the regional road from Dawes to West Agnew.

Dawes is only locally connected to East Agnew. East Agnew is regionally isolated, but there are no other regional paths to East Agnew.

Dawes is regionally connected to Elbridge.

Elbridge is regionally connected to Central Agnew. The shortest regional path from Elbridge to Agnew enters Agnew in the Central Agnew fragment using the border between Burr and Agnew.

Elbridge is only locally connected to East Agnew. East Agnew is regionally isolated, but there are no other regional paths to East Agnew.

The equivalent graph colors the nodes of the districts in different colors. The solid lines count as links. The dashed lines represent local connections between nodes that do not count as links. The red lines indicate links and local connections between nodes in different districts.

Principle: The quality of the shape of a district is based on how well its areas are connected to each other and to the extent that connected areas are split between different districts. Irregular shapes not due to natural barriers and political boundaries at the borders suggest a poor quality district shape. Unrelated areas connected through unusual means suggest a poor quality district shape. A small district in a densely populated area should be measured in a way that is comparable to a large district in a sparsely populated area.

Definition: Cut link. A cut link is a connection between nodes in different districts.

Definition: Erosity. The erosity of a district is the number of cut links to nodes in that district. (nb. Cut links indicate that areas that share transporation interests are in different districts. This reflects on the external quality of a district and correlates to the geographic shape of the district.)

Item E7: The erosity of a plan is the number of cut links in the plan. Mathematically the number of cut links in a plan will equal one half of the sum of the number of cut links for all the districts. This is because each cut link for a district shows up in the count of two districts.

Example 1:

In this example there is a state with nine counties labeled A through I with regional roads shown as heavy black lines. Counties A through D form one district and the rest form a second district.

The equivalent graph replaces roads with links between nodes. The nodes are colored to show which district they are in. Blue links are internal to a district. Red links are cut links between districts.

There are seven cut links in the plan so the erosity is 7. Each district also has an erosity of 7 which is the number of cut links associated with nodes in the district.

Example 2:

This example uses the same underlying map as in the previous example. One district now consists of whole counties B through D plus a fragment of E, shown with an outlined node. The remaining counties and fragment form the other district.

The rules for links to fragments changes the graph in county E. The new graph has only five cut links for an erosity of 5. The chop of E created a more compact shape that was reflected in the lower score. The trade off between chops and erosity is the key element in scoring.

This special rule deals with the issue of nick paths and district that grab a fragment without a local connection.

It is worthwhile reiterating a couple of previous principles with emphasis added.

Principle: Road connections between areas are an indication of a community of interest between those areas. State and federal highways are a stronger indication of a connection than local roads. Large areas at the regional scale of counties should be connected by better roads than small areas at the scale of smaller governmental units.

Principle: The quality of the shape of a district is based on how well its areas are connected to each other and to the extent that connected areas are split between different districts. Irregular shapes not due to natural barriers and political boundaries at the borders suggest a poor quality district shape. Unrelated areas connected through unusual means suggest a poor quality district shape. A small district in a densely populated area should be measured in a way that is comparable to a large district in a sparsely populated area.

Those lead me to this rule.

Definition: Component. A component of a district is a subset of all nodes in the district such that any two nodes of the set are connected to each other by a sequence of links in the district. Each district will consist of one or more disconnected components.

Item E8: Each component in a district in excess of one increases the erosity of the plan by one. (nb. Disconnected components indicate that the district includes areas that are of a regional scale but rely on local not regional connections. This reflects on the internal quality of the district. This item allows districts to be connected by a regional path that nicks another county as long as there is a separate local connection. It is not preferred.)

Example:

This example was used previously to show links and other local paths with isolated fragments. The 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. There are are three districts creating three fragments from Agnew, called West Agnew, Central Agnew, and East Agnew. The nodes for the counties are indicated with solid stars and the nodes for the fragments are shown with hollow stars.

Roads that count for regional connections are shown with heavy brown lines. They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.

This is the equivalent graph that was created from this example. The nodes of the different districts are in different colors. The solid lines count as links. The dashed lines represent local connections between nodes that do not count as links. The red lines indicate links and local connections between nodes in different districts.

The cut links (solid red lines) give the basic erosity of the plan. Here there are eight cut links so the basic plan erosity is 8. Note that the orange node western district has an erosity of 4, the green node eastern district has an erosity of 6, and the black node central district has an erosity of 6. Those sum to 16 which is two times the plan erosity of 8.

The eastern district has two components since Calhoun is only locally connected to West Agnew. One component consists of Calhoun and the other consists of West Agnew. The extra component increases the plan erosity by 1.

The western district also has two components. Dawes and Elbridge are regionally connected and have a link in the district, but East Agnew is only locally connected to each of those counties. The extra component adds another 1 point to the plan erosity.

The central district has only one component consisting of Burr and Central Agnew.

Definition: A nick path is a regional road that primarily links two counties, but goes slightly into a third county along its path. The simple definition would ban the use of a nick path for lack of a regional connection. An exact definition to provide an exception for nick paths proved elusive.

The special rule in the previous post provides for a way to include nick paths use as long as the counties are locally connected. However, the same rule slightly disfavors their use compared to true regional connections.

Nick path example:

This example was used previously to show links and other local paths with isolated fragments. The 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes for the counties are indicated with solid stars.

Roads that count for regional connections are shown with heavy lines (brown and green). They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.

These counties are divided into two districts. The shaded district in the south west consists of just Burr and Calhoun.

The heavy green line represents a major highway that goes between the county seats of Burr and Calhoun. Along the way it cuts across a corner of Agnew, so it is a nick path. It doesn't count as a regional connection between Burr and Calhoun, just like the other examples using this map.

The rule allows the Burr-Calhoun district to exist, because of the local connection. However Burr and Calhoun fall in two different components in that district.

The equivalent graph shows the two districts with different colored nodes in black and orange. It shows the 4 cut links between nodes in different districts in red. The dashed blue line represents the local connection between the two component orange nodes. With the point for the extra component in the southwest district the total erosity for the plan is 5.

Definition: A bridge fragment is a fragment that connects two or more nodes in a component, such that if the bridge fragment is removed from a component then the component would become disconnected into multiple components. We've debated a lot about whether bridge fragments should be allowed or not, and if allowed should they be disfavored. In this component model, bridge fragments are fine if they provide regional links to otherwise disconnected components.

Example:

This example was used previously to show links and other local paths with isolated fragments. The 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes for the counties are indicated with solid stars and the nodes for the fragments are shown with hollow stars.

Roads that count for regional connections are shown with heavy lines (brown and green). They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.

The district line chops Agnew into north and south fragments. Using the connection rule for fragments the connecting path from Agnew to Dawes enters in North Agnew. The connecting paths from Dawes to the other three counties enter in South Agnew.

The effect of the connections is that Burr is connected to South Agnew and Calhoun is connected to South Agnew. Burr and Calhoun are not regionally connected, so without South Agnew they would be in separate components. South Agnew is a bridge fragment that links Burr and Calhoun into a single component.

The equivalent graph shows the two districts with different colored nodes in black and orange. It shows the 4 cut links between nodes in different districts in red. The dashed blue line represents the local connection between the two component orange nodes, but now there are regular links through the South Agnew node. The total erosity for the plan is 4, but there is a chop. Overall the shapes of the districts look fine. This is the type of tradeoff of chops for erosity that makes sense in the system.

Here are two more examples illustrating nick paths and bridge fragments together. The idea is to illustrate them by example without a scoring definition.

As before the 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes for the counties are indicated with solid stars and the nodes for the fragments are shown with hollow stars.

Roads that count for regional connections are shown with heavy lines (brown and green). They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.

Example with a non-connecting fragment:

The south west fragment includes just the nick path in green. Unlike the bridge fragment in the previous post the South Agnew fragment here only has the path from Calhoun to Agnew enter it which includes some of the nick path. Importantly the nick path is not the connecting path from Burr to Agnew, which is the brown highway towards the east. Thus there is a link from Calhoun to South Agnew, but no link from Burr to South Agnew.

The equivalent graph shows the pattern of links as solid lines and local connections as dashed lines. Red links are cut links between nodes in different districts. Burr is in a different component than Calhoun and South Agnew. The graph has four cut links and one extra component for and erosity score of 5.

Example with a connecting fragment:

In this example the connecting path from Agnew to Calhoun has been moved to a separate road, north of the nick path. The nick path now is part of neither the connecting path from Agnew to Burr or Calhoun.

South Agnew is now an isolated fragment. Links a calculated based on the South Agnew node and it has regional connections to both Burr and Calhoun. That allows South Agnew to become a bridge fragment between Burr and Calhoun, putting them in the same component.

Though the extra component is gone, there are now five cut links. The erosity is 5.

One of the options considered during the debate about nick paths was to give some preference when they weren't the path to the node of the intervening county. When the nick path is not the connecting path to another node, it can be used in a connecting bridge fragment in a way that it couldn't when it shared its path.